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Theorem fin1a2lem6 8786
Description: Lemma for fin1a2 8796. Establish that  om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
fin1a2lem.aa  |-  S  =  ( x  e.  On  |->  suc  x )
Assertion
Ref Expression
fin1a2lem6  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )

Proof of Theorem fin1a2lem6
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.aa . . . 4  |-  S  =  ( x  e.  On  |->  suc  x )
21fin1a2lem2 8782 . . 3  |-  S : On
-1-1-> On
3 fin1a2lem.b . . . . 5  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
43fin1a2lem4 8784 . . . 4  |-  E : om
-1-1-> om
5 f1f 5739 . . . 4  |-  ( E : om -1-1-> om  ->  E : om --> om )
6 frn 5695 . . . . 5  |-  ( E : om --> om  ->  ran 
E  C_  om )
7 omsson 6654 . . . . 5  |-  om  C_  On
86, 7syl6ss 3419 . . . 4  |-  ( E : om --> om  ->  ran 
E  C_  On )
94, 5, 8mp2b 10 . . 3  |-  ran  E  C_  On
10 f1ores 5788 . . 3  |-  ( ( S : On -1-1-> On  /\ 
ran  E  C_  On )  ->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
) )
112, 9, 10mp2an 676 . 2  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E )
129sseli 3403 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
b  e.  On )
131fin1a2lem1 8781 . . . . . . . . 9  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
1412, 13syl 17 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( S `  b
)  =  suc  b
)
1514eqeq1d 2430 . . . . . . 7  |-  ( b  e.  ran  E  -> 
( ( S `  b )  =  a  <->  suc  b  =  a
) )
1615rexbiia 2865 . . . . . 6  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  E. b  e.  ran  E  suc  b  =  a )
174, 5, 6mp2b 10 . . . . . . . . . . . 12  |-  ran  E  C_ 
om
1817sseli 3403 . . . . . . . . . . 11  |-  ( b  e.  ran  E  -> 
b  e.  om )
19 peano2 6671 . . . . . . . . . . 11  |-  ( b  e.  om  ->  suc  b  e.  om )
2018, 19syl 17 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  suc  b  e.  om )
213fin1a2lem5 8785 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  e.  ran  E  <->  -. 
suc  b  e.  ran  E ) )
2221biimpd 210 . . . . . . . . . . 11  |-  ( b  e.  om  ->  (
b  e.  ran  E  ->  -.  suc  b  e. 
ran  E ) )
2318, 22mpcom 37 . . . . . . . . . 10  |-  ( b  e.  ran  E  ->  -.  suc  b  e.  ran  E )
2420, 23jca 534 . . . . . . . . 9  |-  ( b  e.  ran  E  -> 
( suc  b  e.  om 
/\  -.  suc  b  e. 
ran  E ) )
25 eleq1 2494 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( suc  b  e.  om  <->  a  e.  om ) )
26 eleq1 2494 . . . . . . . . . . 11  |-  ( suc  b  =  a  -> 
( suc  b  e.  ran  E  <->  a  e.  ran  E ) )
2726notbid 295 . . . . . . . . . 10  |-  ( suc  b  =  a  -> 
( -.  suc  b  e.  ran  E  <->  -.  a  e.  ran  E ) )
2825, 27anbi12d 715 . . . . . . . . 9  |-  ( suc  b  =  a  -> 
( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  <-> 
( a  e.  om  /\ 
-.  a  e.  ran  E ) ) )
2924, 28syl5ibcom 223 . . . . . . . 8  |-  ( b  e.  ran  E  -> 
( suc  b  =  a  ->  ( a  e. 
om  /\  -.  a  e.  ran  E ) ) )
3029rexlimiv 2850 . . . . . . 7  |-  ( E. b  e.  ran  E  suc  b  =  a  ->  ( a  e.  om  /\ 
-.  a  e.  ran  E ) )
31 peano1 6670 . . . . . . . . . . . . . 14  |-  (/)  e.  om
323fin1a2lem3 8783 . . . . . . . . . . . . . 14  |-  ( (/)  e.  om  ->  ( E `  (/) )  =  ( 2o  .o  (/) ) )
3331, 32ax-mp 5 . . . . . . . . . . . . 13  |-  ( E `
 (/) )  =  ( 2o  .o  (/) )
34 om0x 7176 . . . . . . . . . . . . 13  |-  ( 2o 
.o  (/) )  =  (/)
3533, 34eqtri 2450 . . . . . . . . . . . 12  |-  ( E `
 (/) )  =  (/)
36 f1fun 5741 . . . . . . . . . . . . . 14  |-  ( E : om -1-1-> om  ->  Fun 
E )
374, 36ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  E
38 f1dm 5743 . . . . . . . . . . . . . . 15  |-  ( E : om -1-1-> om  ->  dom 
E  =  om )
394, 38ax-mp 5 . . . . . . . . . . . . . 14  |-  dom  E  =  om
4031, 39eleqtrri 2505 . . . . . . . . . . . . 13  |-  (/)  e.  dom  E
41 fvelrn 5974 . . . . . . . . . . . . 13  |-  ( ( Fun  E  /\  (/)  e.  dom  E )  ->  ( E `  (/) )  e.  ran  E )
4237, 40, 41mp2an 676 . . . . . . . . . . . 12  |-  ( E `
 (/) )  e.  ran  E
4335, 42eqeltrri 2503 . . . . . . . . . . 11  |-  (/)  e.  ran  E
44 eleq1 2494 . . . . . . . . . . 11  |-  ( a  =  (/)  ->  ( a  e.  ran  E  <->  (/)  e.  ran  E ) )
4543, 44mpbiri 236 . . . . . . . . . 10  |-  ( a  =  (/)  ->  a  e. 
ran  E )
4645necon3bi 2627 . . . . . . . . 9  |-  ( -.  a  e.  ran  E  ->  a  =/=  (/) )
47 nnsuc 6667 . . . . . . . . 9  |-  ( ( a  e.  om  /\  a  =/=  (/) )  ->  E. b  e.  om  a  =  suc  b )
4846, 47sylan2 476 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  om  a  =  suc  b )
49 eleq1 2494 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( a  e.  om  <->  suc  b  e.  om )
)
50 eleq1 2494 . . . . . . . . . . . . . . . . 17  |-  ( a  =  suc  b  -> 
( a  e.  ran  E  <->  suc  b  e.  ran  E ) )
5150notbid 295 . . . . . . . . . . . . . . . 16  |-  ( a  =  suc  b  -> 
( -.  a  e. 
ran  E  <->  -.  suc  b  e. 
ran  E ) )
5249, 51anbi12d 715 . . . . . . . . . . . . . . 15  |-  ( a  =  suc  b  -> 
( ( a  e. 
om  /\  -.  a  e.  ran  E )  <->  ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E ) ) )
5352anbi1d 709 . . . . . . . . . . . . . 14  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om ) 
<->  ( ( suc  b  e.  om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om ) ) )
54 simplr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  -.  suc  b  e. 
ran  E )
5521adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  ( b  e.  ran  E  <->  -.  suc  b  e.  ran  E ) )
5654, 55mpbird 235 . . . . . . . . . . . . . 14  |-  ( ( ( suc  b  e. 
om  /\  -.  suc  b  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E
)
5753, 56syl6bi 231 . . . . . . . . . . . . 13  |-  ( a  =  suc  b  -> 
( ( ( a  e.  om  /\  -.  a  e.  ran  E )  /\  b  e.  om )  ->  b  e.  ran  E ) )
5857com12 32 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  b  e. 
om )  ->  (
a  =  suc  b  ->  b  e.  ran  E
) )
5958impr 623 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  b  e.  ran  E )
60 simprr 764 . . . . . . . . . . . 12  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  a  =  suc  b )
6160eqcomd 2434 . . . . . . . . . . 11  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  suc  b  =  a )
6259, 61jca 534 . . . . . . . . . 10  |-  ( ( ( a  e.  om  /\ 
-.  a  e.  ran  E )  /\  ( b  e.  om  /\  a  =  suc  b ) )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) )
6362ex 435 . . . . . . . . 9  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( (
b  e.  om  /\  a  =  suc  b )  ->  ( b  e. 
ran  E  /\  suc  b  =  a ) ) )
6463reximdv2 2835 . . . . . . . 8  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  ( E. b  e.  om  a  =  suc  b  ->  E. b  e.  ran  E  suc  b  =  a ) )
6548, 64mpd 15 . . . . . . 7  |-  ( ( a  e.  om  /\  -.  a  e.  ran  E )  ->  E. b  e.  ran  E  suc  b  =  a )
6630, 65impbii 190 . . . . . 6  |-  ( E. b  e.  ran  E  suc  b  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
6716, 66bitri 252 . . . . 5  |-  ( E. b  e.  ran  E
( S `  b
)  =  a  <->  ( a  e.  om  /\  -.  a  e.  ran  E ) )
68 f1fn 5740 . . . . . . 7  |-  ( S : On -1-1-> On  ->  S  Fn  On )
692, 68ax-mp 5 . . . . . 6  |-  S  Fn  On
70 fvelimab 5881 . . . . . 6  |-  ( ( S  Fn  On  /\  ran  E  C_  On )  ->  ( a  e.  ( S " ran  E
)  <->  E. b  e.  ran  E ( S `  b
)  =  a ) )
7169, 9, 70mp2an 676 . . . . 5  |-  ( a  e.  ( S " ran  E )  <->  E. b  e.  ran  E ( S `
 b )  =  a )
72 eldif 3389 . . . . 5  |-  ( a  e.  ( om  \  ran  E )  <->  ( a  e. 
om  /\  -.  a  e.  ran  E ) )
7367, 71, 723bitr4i 280 . . . 4  |-  ( a  e.  ( S " ran  E )  <->  a  e.  ( om  \  ran  E
) )
7473eqriv 2425 . . 3  |-  ( S
" ran  E )  =  ( om  \  ran  E )
75 f1oeq3 5767 . . 3  |-  ( ( S " ran  E
)  =  ( om 
\  ran  E )  ->  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S " ran  E
)  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E
) ) )
7674, 75ax-mp 5 . 2  |-  ( ( S  |`  ran  E ) : ran  E -1-1-onto-> ( S
" ran  E )  <->  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om 
\  ran  E )
)
7711, 76mpbi 211 1  |-  ( S  |`  ran  E ) : ran  E -1-1-onto-> ( om  \  ran  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   E.wrex 2715    \ cdif 3376    C_ wss 3379   (/)c0 3704    |-> cmpt 4425   dom cdm 4796   ran crn 4797    |` cres 4798   "cima 4799   Oncon0 5385   suc csuc 5387   Fun wfun 5538    Fn wfn 5539   -->wf 5540   -1-1->wf1 5541   -1-1-onto->wf1o 5543   ` cfv 5544  (class class class)co 6249   omcom 6650   2oc2o 7131    .o comu 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-omul 7142
This theorem is referenced by:  fin1a2lem7  8787
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